Infinitely many zero-Pisano-Legendre primes conjecture

Let $a$ and $b$ be integers with $(a,b)\neq(0,0)$. Let $\chi(a,b)$ be the set of admissible indices, and for $k\in\chi(a,b)$ define

$$\vartheta_{(a,b)}(k)=\{p:p\text{ is a }k\text{-Pisano-Legendre prime relative to }(a,b)\}.$$

Infinitely many zero-Pisano-Legendre primes conjecture. If $0\in\chi(a,b)$, then

$$|\vartheta_{(a,b)}(0)|=\aleph_0.$$

The claim predicts infinitely many zero-Pisano-Legendre primes whenever zero is an admissible index. The source presents increasing numerical counts as motivation, but gives no proof or resolution.